Problem 17 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 6 and 15

Step-by-Step Solution

Answer

The least common multiple of 6 and 15 is 30.

Step 1: Find Prime Factors

First, we need to find the prime factorization of each number. The prime factorization of 6 is \(2 \times 3\). The prime factorization of 15 is \(3 \times 5\).

Step 2: Identify Highest Powers of Each Prime

Next, identify the highest power of each prime number present in either factorization. For the number 2, the highest power is \(2^1\) (from 6). For the number 3, the highest power is \(3^1\) (from both 6 and 15). For the number 5, the highest power is \(5^1\) (from 15).

Step 3: Calculate the Least Common Multiple

Multiply the highest powers of each prime number to find the least common multiple (LCM). So, the LCM is \(2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30\).

Key Concepts

Prime Factorization

Prime factorization is a process of breaking down a number into a product of prime numbers. A prime number is one that is greater than 1 and has no divisors other than 1 and itself. For example, the number 6 can be broken down into the prime numbers 2 and 3. Hence, its prime factorization is expressed as \(2 \times 3\). Similarly, the number 15 is also broken into its prime factors, which are 3 and 5. Therefore, its prime factorization is \(3 \times 5\).
Prime factorization is important as it helps in finding commonality between numbers. It lays the foundation for determining the least common multiple or greatest common divisor of a set of numbers. Understanding this concept can simplify calculations in algebra and arithmetic.
Learning to perform prime factorization requires practice. It is helpful to memorize the list of prime numbers up to a certain limit, like 30, to expedite the factorization process. Often, smaller numbers can be split into prime numbers easily, making calculations faster.

Highest Power

When finding the least common multiple (LCM) of two or more numbers, identifying the highest power of each prime is essential. When we say "highest power," it refers to the most occurrences of a particular prime number in any of the original prime factorizations. For instance, in our problem, the number 2 appears as \(2^1\) only in the factorization of 6.
For prime number 3, it appears as \(3^1\) in both numbers (6 and 15), but we still only consider \(3^1\), because it's the highest power in either factorization. As for number 5, it shows up as \(5^1\) in the factorization of 15 and not at all in 6. Thus, for each prime present, we take the highest power from among the factorizations.
This concept ensures that when the LCM is found, it is a multiple of each original number, accommodating all prime contributions in their maximum required amount. It essentially binds together the prime factors accounting for the largest necessity of each.

Multiplication of Primes

Multiplication of the highest powers of primes is the final step in determining the least common multiple. Once you have identified the highest power of each prime from the factorizations, you simply multiply these together to find the LCM.
For instance, using the highest powers from our example:

• \(2^1 = 2\)
• \(3^1 = 3\)
• \(5^1 = 5\)

When you multiply these results together, you get \(2 \times 3 \times 5 = 30\). This product, 30, is the smallest number that is a multiple of both 6 and 15.
This method of finding the LCM ensures that all prime factors are included the proper number of times. It guarantees that the result is the smallest possible multiple that can evenly divide by each of the original numbers. It is a straightforward process once you master the concept of prime factorization and calculating the highest power of primes.